The evolution of closed curves under (REF ) has received considerable study. For example, Gage ([1]}), Gage-Hamilton ([2]}) and
Grayson ([3]}), considered the evolution of the convex curve flowed by the mean curvature in the plane. Furthermore, Grayson ([4]}) generalized the results into the case on the embedded closed curve of Riemannian surfaces. Huisken ([5]}) and Andrews-Bryan ([6]}) used the technique of distance comparison to investigate the curve shortening flow on the plane. For the mean curvature flow of the submanifolds with codimension \(k\ge 2\) , Andrews-Baker ([7]}) proposed a machinery about the evolution of the second fundamental form \(A\)  with arbitrary codimension. They also obtained the convergence of the mean curvature flow for the submanifold pinched to sphere in Euclidean space. More recently, with an integral curvature condition ([8]}) and a pinched condition in hyperbolic space form ([9]}) Liu, Xu, Ye and Zhao also proved two extension theorems for the mean curvature flow of the submanifold with high codimension. Smoczyk's survey ([10]}) is a good reference.
